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Theorem fnres 5199
 Description: An equivalence for functionality of a restriction. Compare dffun8 5134. (Contributed by Mario Carneiro, 20-May-2015.)
Assertion
Ref Expression
fnres ((F A) Fn Ax A ∃!y xFy)
Distinct variable groups:   x,y,A   x,F,y

Proof of Theorem fnres
StepHypRef Expression
1 ancom 437 . . 3 ((x A ∃*y xFy x A y xFy) ↔ (x A y xFy x A ∃*y xFy))
2 brres 4949 . . . . . . . . 9 (x(F A)y ↔ (xFy x A))
3 ancom 437 . . . . . . . . 9 ((xFy x A) ↔ (x A xFy))
42, 3bitri 240 . . . . . . . 8 (x(F A)y ↔ (x A xFy))
54mobii 2240 . . . . . . 7 (∃*y x(F A)y∃*y(x A xFy))
6 moanimv 2262 . . . . . . 7 (∃*y(x A xFy) ↔ (x A∃*y xFy))
75, 6bitri 240 . . . . . 6 (∃*y x(F A)y ↔ (x A∃*y xFy))
87albii 1566 . . . . 5 (x∃*y x(F A)yx(x A∃*y xFy))
9 dffun6 5124 . . . . 5 (Fun (F A) ↔ x∃*y x(F A)y)
10 df-ral 2619 . . . . 5 (x A ∃*y xFyx(x A∃*y xFy))
118, 9, 103bitr4i 268 . . . 4 (Fun (F A) ↔ x A ∃*y xFy)
12 dmres 4986 . . . . . . 7 dom (F A) = (A ∩ dom F)
13 inss1 3475 . . . . . . 7 (A ∩ dom F) A
1412, 13eqsstri 3301 . . . . . 6 dom (F A) A
15 eqss 3287 . . . . . 6 (dom (F A) = A ↔ (dom (F A) A A dom (F A)))
1614, 15mpbiran 884 . . . . 5 (dom (F A) = AA dom (F A))
17 dfss3 3263 . . . . 5 (A dom (F A) ↔ x A x dom (F A))
1812elin2 3446 . . . . . . . 8 (x dom (F A) ↔ (x A x dom F))
1918baib 871 . . . . . . 7 (x A → (x dom (F A) ↔ x dom F))
20 eldm 4898 . . . . . . 7 (x dom Fy xFy)
2119, 20syl6bb 252 . . . . . 6 (x A → (x dom (F A) ↔ y xFy))
2221ralbiia 2646 . . . . 5 (x A x dom (F A) ↔ x A y xFy)
2316, 17, 223bitri 262 . . . 4 (dom (F A) = Ax A y xFy)
2411, 23anbi12i 678 . . 3 ((Fun (F A) dom (F A) = A) ↔ (x A ∃*y xFy x A y xFy))
25 r19.26 2746 . . 3 (x A (y xFy ∃*y xFy) ↔ (x A y xFy x A ∃*y xFy))
261, 24, 253bitr4i 268 . 2 ((Fun (F A) dom (F A) = A) ↔ x A (y xFy ∃*y xFy))
27 df-fn 4790 . 2 ((F A) Fn A ↔ (Fun (F A) dom (F A) = A))
28 eu5 2242 . . 3 (∃!y xFy ↔ (y xFy ∃*y xFy))
2928ralbii 2638 . 2 (x A ∃!y xFyx A (y xFy ∃*y xFy))
3026, 27, 293bitr4i 268 1 ((F A) Fn Ax A ∃!y xFy)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∀wral 2614   ∩ cin 3208   ⊆ wss 3257   class class class wbr 4639  dom cdm 4772   ↾ cres 4774  Fun wfun 4775   Fn wfn 4776 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790 This theorem is referenced by: (None)
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